By Kollar J., Lazarsfeld R., Morrison D. (eds.)

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**Additional info for Algebraic Geometry Santa Cruz 1995, Part 2**

**Example text**

3) Let g be of order m and let fm be the number of fixed points of g. Then fm is given in the following Table 3. Table 3: The number of fixed points of finite symplectic automorphisms of K3 surfaces m2345678 fm 8 6 4 4 2 3 2 Recall that the Mathieu group M24 acts on the set Ω = {1, . . , 24} of 24 letters. Let M23 be the stabilizer subgroup of the letter 1. Then M23 is also a finite sporadic simple group, called the Mathieu group of degree 23. The conjugacy classes of M23 are determined by their orders and are given in the following Table 4.

3. H i (X, C) F r H i (X, C) F d−r+1 H 2d−i (X, C)∨. 5. Let A ⊂ R be a subring. An A-Hodge structure (HS) of weight N ∈ Z is given by the following datum: • A finitely generated A-module V, and either of the two equivalent statements below: •1 A decomposition VC = V p,q , V p,q = V q,p , p+q=N where − is complex conjugation induced from conjugation on the second factor C of VC := V ⊗ C. •2 A finite descending filtration VC ⊃ · · · ⊃ F r ⊃ F r−1 ⊃ · · · ⊃ {0}, satisfying VC = F r F N−r+1 , ∀ r ∈ Z.

J. Math. 12, 191–282 (1986) 21. S. Kond¯o, The rationality of the moduli space of Enriques surfaces. Compos. Math. 91, 159–173 (1994) 22. S. Kond¯o, Niemeier lattices, Mathieu groups, and finite groups of symplectic automorphisms of K3 surfaces (with an appendix by S. Mukai). Duke Math. J. 92, 593–603 (1998) 23. S. Kond¯o, The automorphism group of a generic Jacobian Kummer surface. J. Algebr. Geom. 7, 589–609 (1998) 24. S. Kond¯o, On the Kodaira dimension of the moduli space of K3 surfaces II.